# Full text of "Treatise On Analysis Vol-Ii"

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```268       XIV   INTEGRATION IN LOCALLY COMPACT GROUPS
Let /be any function belonging to tf(G), and put

h(jf) = f /(*{) da'«)       for   n(x) = x".
By virtue of (14.4.2) we have

[f(u-\x))d*(x) = (d^(±)(f(u-\x)u-\^d^(^
and since u"1^) = u"~l(£) by definition,

f /(iT'OOiT1®) <fa'«) = (mod(iO) ' f /(""'WO <**'(«)

^(w-'Oc)).

Since J h(ulf'1(xir)) da"(x") = (mod(w")) J *(*") da,"(x"\ we obtain the required
result.

(1 4.4.7)   For each xeG we have

where i* denotes the automorphism sf\-*x~ls'x of G'.

Apply (14.4.6) with u = ix9 and use (14.3.7). In particular, if x' e G', we
have AG(jcO = AG,(^).

PROBLEMS

1.    Let G be a locally compact group. A measure jn on G is said to be relatively left-
(resp. right-) invariant if for each s e G there exists a complex number x(\$) ^ 0 such
that y(s)[ji = xWp fresP- S(^)/x = xC5)^)- Show that x is a continuous homomorphism
of G into the multiplicative group C*, and that ^ is of the form a*£~l • ft, where fi is a
left (resp. right) Haar measure on G and a is a complex number =^0 (Section 14.3,
Problem 4(a)). Consider the converse of this result, and deduce that every relatively
left-invariant measure on G is also relatively right-invariant.

2.    Let G be a locally compact group, H a closed subgroup of G, a a left Haar measure on
G, £ a left Haar measure on H, TT the canonical mapping of G onto G/H.

(a)    For each function / e Jf (G), show that there exists a unique function / fr e Jf (G/H)

such that /b(7r(x)) = f /(*£) </£(£). Conversely, for each A e ^f (G/H), there exists a

JH

function/e «#XG) such that/I7 = /z. (If C is the support of A, consider a compact subset
K of G such that 7r(K) — C (12.10.9) and a function # e ^T(G) such that #(#) > 0 for
all x e K, and observe that (g • (h ° ?r))b — g* - h.

(b)   A measure /x ^ 0 on G/H is said to be relatively invariant under G if for eachG). Then modG(w) =
```